Eulerian polynomials of spherical type

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Eulerian polynomials of spherical type

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Cohen, A. M. (2008). Eulerian polynomials of spherical type. Münster Journal of Mathematics, 1, 1-8.

Document status and date: Published: 01/01/2008

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urn:nbn:de:hbz:6-43529471561 by the author 2008c

Eulerian polynomials of spherical type

Arjeh M. Cohen

(Communicated by Linus Kramer)

Abstract. The Eulerian polynomial of a finite Coxeter system (W, S) of rank n records, for each k ∈ {1, . . . , n}, the number of elements w ∈ W with an ascent set {s ∈ S | l(ws) > l(w)} of size k, where l(w) denotes the length of w with respect to S. The classical Eulerian polynomial occurs when the Coxeter group has type An, so W is the symmetric group on

n+ 1 letters. Victor Reiner gave a formula for arbitrary Eulerian polynomials and showed how to compute them in the classical cases. In this note, we compute the Eulerian polynomial for any spherical type.

Let M be a Coxeter matrix of rank n. This means M is a symmetric n × n matrix with entries in N such that Mii= 1 and Mij> 1 if i 6= j. We also refer

to M as a diagram, that is, an edge-labeled graph with nodes {1, . . . , n} and edge {i, j} labeled Mij whenever Mij > 2. Our setting will involve Coxeter

groups as introduced in [3]. Accordingly, we let (W, S) be a Coxeter system of type M . Then {1, . . . , n} and the set S = {s1, . . . , sn} of simple reflections

are in bijective correspondence and we will often identify the two, so S can be viewed as the set of nodes of M . We also write W (M ) instead of W to record the dependence on M . If W (M ) is finite, then M is called spherical. The connected spherical diagrams M are An (n ≥ 1), Bn (n ≥ 2), Dn (n ≥ 4), En

(n = 6, 7, 8), F4, G2, Hn (n = 3, 4), and I(m)₂ (m ≥ 3). The double occurrences

in this list are A2= I(3)2 , B2= I(4)2 , and G2= I(6)2 . The nodes of these diagrams

are labeled as in [3]. In this note, we assume that M is spherical.

Two great assets of the study of Coxeter groups are the reflection representa-tion ρ and the root system Φ. Both are related to the vector space V = ⊕iRαi

with formal basis αi (1 ≤ i ≤ n) supplied with the symmetric bilinear form

(·, ·) determined by

(αi, αj) = −2 cos(2π/Mij)

for 1 ≤ i, j ≤ n. The reflection representation of W is the group homomor-phism ρ from W to the orthogonal group on V with respect to (·, ·) for which

ρ(s)αj = αj− (αj, αs)αs,

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2 Arjeh M. Cohen

As M is spherical, (·, ·) is positive definite, so W may be viewed as a finite real orthogonal group in n dimensions. Now Φ =S

s∈SW αs is a root system

in the sense of [6]; its members are called roots. The elements αsfor s ∈ S are

called the simple roots. In the case of a Weyl group, a root system in the sense of [3] can be obtained from Φ by adjusting the length of certain roots. The set of positive roots of Φ is defined to be Φ+_{= Φ ∩ (⊕}

sR≥0αs). It is well known

that Φ is the disjoint union of Φ+ _{and −Φ}+_.

For j ∈ {1, . . . , n}, define pj to be the number of elements w ∈ W such

that {s ∈ S | ρ(w)αs∈ Φ+} has size j. This number is related to the descent

statistics discussed in [2, 10]. The Eulerian polynomial of type M is P (M, t) =

n

X

j=0

pjtj.

If M = An, then, W ∼= Σn+1, the symmetric group on n + 1 letters, and,

as a W -set, Φ can be identified with the set of distinct ordered pairs (i, j) for 1 ≤ i, j ≤ n + 1 in such a way that the simple roots are the pairs (i, i + 1) for 1 ≤ i ≤ n, and the positive roots are all (i, j) with i < j. In this case, pi is the

number of π ∈ Σn+1such that π(i) < π(i + 1).

The coefficient of ti _{in the polynomial P (M, 1 + t) equals the number of}

i-dimensional faces of the polytope (permutahedron) associated to M . The corresponding toric variety has only even-dimensional Betti numbers; these are the coefficients of P (M, t). The signature of the toric variety equals P (M, −1); see [8, 1].

I am grateful to Prof. Hirzebruch for drawing my attention to this poly-nomial and his inspiring lecture at the Killing meeting in M¨unster, December 7, 2007, where he posed the problem of computing the Eulerian polynomial for M = E8. The results for Coxeter groups of classical types are known and

appear in Theorem 4 below; the results for the exceptional spherical types are given in Table 1. We will derive all of these results from the following expres-sion for the Eulerian polynomial in terms of standard parabolic subgroups of W . Here a standard parabolic subgroup of W is a subgroup WJ generated by

a subset J of S. The formula is a special case of [9, Theorem 1], of which we give a proof that does not essentially differ from the original.

Proposition 1. The Eulerian polynomial for spherical type M is determined by P (M, t) = X K⊆S |W |(t − 1)|K| |WK| , where (W, S) is the Coxeter system of type M .

Proof. For J ⊆ S, define pJ to be the number of elements w ∈ W such that

{s ∈ S | ρ(w)αs∈ Φ+} = J. For w ∈ W , let l(w) be the minimum length q of

an expression of w as a product r1· · · rqof members riof S. The proof is based

on two facts, which are well known in Coxeter group theory (cf. [3, 5, 7]). The first is the fact that ρ(w)αs∈ Φ+is equivalent to l(ws) > l(w). The second is

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the fact that, for given J ⊆ S, the set of elements w ∈ W with l(ws) > l(w) for each s ∈ J is a complete set of distinguished coset representatives of WJ,

the subgroup of W generated by all members of J. In particular, its size is |W/WJ| and so, by inclusion/exclusion,

pJ = X K⊇J (−1)|K|+|J||W/WK| for each J ⊆ S. As a consequence, P (M, t) = X J⊆S pJt|J|= X J⊆S X K⊇J (−1)|K|+|J||W/WK|t|J| = X K⊆S (−1)|K||W/WK| X J⊆K (−t)|J| = X K⊆S (−1)|K||W/WK|(1 − t)|K| = X K⊆S |W/WK|(t − 1)|K|. Here are some immediate observations on these polynomials.

Lemma 2. The Eulerian polynomial P (M, t) satisfies the following properties. (i) P (M, 1) = |W | and p0= 1.

(ii) P (A1, t) = 1 + t.

(iii) P (I(m)2 , t) = 1 + (2m − 2)t + t2.

(iv) If M is the disjoint and disconnected union of the diagrams M1 and M2

then P (M, t) = P (M1, t)P (M2, t).

(v) P (M, t) = tn_{P (M, t}−1_).

Proof. (i) is clear from the definition. (ii) and (iii) follow directly from Propo-sition 1. (iv) follows from the decompoPropo-sitions W (M ) = W (M1) × W (M2) and

Φ = Φ1 ˙∪ Φ2. (v) is equivalent to pj = pn−j for each j ∈ {0, . . . , n}, which

follows from left multiplication by the longest element w0. For, ρ(w)αs∈ Φ+

if and only if ρ(w0w)αs ∈ −Φ+, so, for each J ⊆ S, left multiplication by

w0 gives a bijection between {w ∈ W | {s ∈ S | ρ(w)αs ∈ Φ+} = J} and

{w ∈ W | {s ∈ S | ρ(w)αs∈ Φ+} = S \ J}, proving pJ = pS\J.

If we apply Proposition 1 directly, we need to consider all 2n _subdiagrams

of M . The following corollary of Proposition 1 reduces that number to the number of connected components on a given node of M . Fix k ∈ S and let K(M, k) denote the collection of the empty set and the connected subsets of S containing k. For I ∈ K(M, k), write N (I) for the set of elements of S equal to or connected with a member of I, with the understanding that N (∅) = {k}. For J ⊆ S, we write M \ J to denote the diagram induced by M on S \ J.

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4 Arjeh M. Cohen

Corollary 3. Let M be a spherical Coxeter diagram and (W, S) a Coxeter system. Then, for each k ∈ S,

P (M, t) = X

I∈K(M,k)

|W |(t − 1)|I|

|WI| · |WS\N (I)|

P (M \ N (I), t).

Proof. Using Proposition 1, we first sum over the connected components I of a subdiagram containing the node k and next over possible completions of I to a subset J of S. If I is a nonempty member of K(M, k), then, taking the sum over all subsets of S of which I is a connected component yields precisely the summand for I in the sum in the corollary. For I = ∅, the possible completions are the subsets of S \ {k}. This explains the choice N (∅) = {k}

and the summand for that value of N .

The following result gives efficient recursion formulas for the classical Weyl groups. Part (iii) is due to Stembridge; cf. [10, Section 4].

Theorem 4. The polynomials P (M, t) for M one of An (n ≥ 1), Bn (n ≥ 2),

Dn (n ≥ 4) satisfy the following recursion, where P (A−1, t) = P (A0, t) = 1.

P (An, t) = n X i=0 n + 1 i + 1 P (An−i−1, t)(t − 1)i. P (Bn, t) = n X i=0 2n−in i P (An−i−1, t)(t − 1)i. P (Dn, t) = P (Bn, t) − 2n−1ntP (An−2, t)

Proof. We apply Corollary 3. In all cases, the summation index i equals the size of the connected component in K(M, n) whose induced subdiagram in M is a straight path starting at n directed towards 1; it has type Ai. For An, this

gives P (An, t) = n X i=0 |W (An)|(t − 1)i |W (Ai)| · |W (An−i−1)| P (An−i−1, t) = n X i=0 n + 1 i + 1 P (An−i−1, t)(t − 1)i. For Bn, we find P (Bn, t) = n X i=0 |W (Bn)|(t − 1)i |W (Bi)| · |W (An−i−1)| P (An−i−1, t) = n X i=0 2n−in i P (An−i−1, t)(t − 1)i.

As for Dn, we set aside the members ∅ and {n} of K(Dn, n); the corresponding

summands appear separately in the summation below. The summation index j (3 ≤ j ≤ n) equals the size of the component in K(Dn, n) distinct from the

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straight path towards 1; it contains n − 2 and n − 1 and has type Dj. Of

Lemma 2, we use (ii) for P (A1, t) and (iv) for P (An−3A1, t).

P (Dn, t) = |W (Dn)| |W (An−1)|P (An−1, t) + |W (Dn)|(t − 1) |W (A1)| · |W (An−3A1)|P (An−3A1, t) + n−1 X i=2 |W (Dn)|(t − 1)i |W (Ai)| · |W (An−i−2)|P (An−i−2, t) + n X j=3 |W (Dn)|(t − 1)j |W (Dj)| · |W (An−j−1)| P (An−j−1, t) = 2n−1P (An−1, t) + 2n−2n 2 (t2− 1)P (An−3, t) + n−1 X i=2 2n−1 _n i + 1 (t − 1)i_{P (A} n−i−2, t) + n X j=3 2n−jn j (t − 1)j_{P (A} n−j−1, t)

We equate the two summations as P (An−1, t) and P (Bn, t) up to some scalars

and a few missing terms, as follows. By compensating for the i = 0 and i = 1 terms, we find that the first summation, over i, contributes

−2n−1nP (An−2, t) − 2n−1

n 2

(t − 1)p(An−3, t) + 2n−1P (An−1, t),

and, by compensating for j = 0, 1, 2, we find that the last summation, over j, contributes − 2nP (An−1, t) − 2n−1n(t − 1)P (An−2, t) − 2n−2n 2 (t − 1)2_{P (A} n−3, t) + P (Bn, t).

Substituting these contributions for the summations in the above expression for P (Dn, t), we find P (Dn, t) = 2n−1P (An−1, t) + 2n−2 n 2 (t2− 1)P (An−3, t) − 2n−1_{nP (A} n−2, t) − 2n−1 n 2 (t − 1)p(An−3, t) + 2n−1_{P (A} n−1, t) − 2nP (An−1, t) − 2n−1n(t − 1)P (An−2, t) − 2n−2n 2 (t − 1)2P (An−3, t) + P (Bn, t) = −2n−1ntP (An−2, t) + P (Bn, t).

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6 Arjeh M. Cohen

Table 1. Eulerian polynomials of non-classical connected types M P (M, t) H3 1 + 59t + 59t2+ t3 F4 1 + 236t + 678t2+ 236t3+ t4 H4 1 + 1316t + 4566t2+ 1316t3+ t4 E6 1 + 1272t + 12183t2+ 24928t3+ 12183t4+ 1272t5+ t6 E7 1 + 17635t + 309969t2+ 1123915t3+ 1123915t4+ 309969t5 + 17635t6_{+ t}7 E8 1 + 881752t + 28336348t2+ 169022824t3+ 300247750t4 + 169022824t5_{+ 28336348t}6_{+ 881752t}7_{+ t}8

As G2 = I(6)2 has been dealt with by Lemma 2(iii), it remains to consider

the Coxeter diagram on non-classical types and rank at least 3.

Theorem 5. For connected non-classical Coxeter diagrams M of rank n ≥ 3, the polynomials P (M, t) are as in Table 1.

Proof. This follows from a systematic application of Corollary 3. For instance, for E6, with k = 6, we find nine components in K(E6, 6), and compute

P (E6, t) = 51840 1920P (D5, t) + 51840 240 P (A4, t)(t − 1) +51840 72 P (A1A2, t)(t − 1) 2₊51840 48 P (A1, t)(t − 1) 3 +51840 120 (t − 1) 4₊51840 720 (t − 1) 5 +51840 240 P (A1, t)(t − 1) 4₊51840 1920 (t − 1) 5₊51840 51840(t − 1) 6 = 1 + 1272t + 12183t2+ 24928t3+ 12183t4+ 1272t5+ t6. The polynomial P (H3, t) can also be determined directly from Lemma 2(i), (v).

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Received February 2, 2008; accepted July 17, 2008

Arjeh M. Cohen

Department of Mathematics and Computer Science Technische Universiteit Eindhoven

P.O. Box 513, 5600 MB Eindhoven, Netherlands E-mail: amc@win.tue.nl

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